A New Insight into Electron Acceleration Properties from Theoretical Modeling of Double-peaked Radio Light Curves in Core-collapse Supernovae

Radio emission from a core-collapse supernova (radio SN) is believed to originate from the interaction between an SN shock and the circumstellar medium (CSM; e.g., Chevalier 1982a, 1998, Chevalier & Fransson 2006; Fransson & Björnsson 1998; Chevalier et al. 2006; Maeda 2012, 2013a; Matsuoka et al. 2019; Matsuoka & Maeda 2020). The properties of radio SNe depend on the density structure of the CSM, and thus we can use them as an indicator of mass-loss histories of SN progenitors (e.g., Chevalier et al. 2006; Chevalier & Fransson 2006; Salas et al. 2013; Margutti et al. 2017; Maeda et al. 2021). Furthermore, modeling radio SNe has a potential to probe the efficiency of the particle acceleration and magnetic field amplification driven by a nonrelativistic shock wave (Maeda 2012; Murase et al. 2014, 2019; Xu & Lazarian 2022). Hence, investigating the nature of radio SNe affords us clues to probing the stellar evolution of massive stars, as well as plasma physics related to the shock acceleration mechanism.

In the classical modeling of radio SNe, the peak luminosity has been determined by the moment when the optical depth to synchrotron self-absorption becomes unity (Chevalier 1998). Then the light curve is characterized by a single peak. In fact, this idea has been successful in fitting radio SNe, including SN 1993J and SN 2011dh, on the assumption of a (nearly) steady wind CSM configuration (see Figure 1 and, e.g., Fransson & Björnsson 1998; Maeda 2012; Soderberg et al. 2012; Chevalier & Fransson 2017).

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That said, there are several objects exhibiting multiple peaks in the radio light curve (see Figure 1). Examples are broad-lined Type Ic SNe 1998bw and 2007bg (Kulkarni et al. 1998; Salas et al. 2013). Another object is SN 2014C, which exhibits the rebrightening of the radio emission a few years after the explosion (see also Stockdale et al. 2009, Mauerhan et al. 2018, and Balasubramanian et al. 2021 for SN 2004dk, and Maeda et al. 2023b for SN 2018ivc). Furthermore, SN 2001em and SN 2017ens show luminous radio emissions in the late phase, albeit the detections of radio emissions in the early phase were not reported (Chandra et al. 2020; Margutti et al. 2023). The boosting of the ejecta velocity has been proposed for SN 1998bw (Li & Chevalier 1999), while multiple CSM components including a shell-like structure are preferred for the other objects (see the above references). We note that the existence of a shell-like CSM component can be an intriguing tracer of the eruptive mass loss of SN progenitors (Margutti et al. 2017).

In this study, we present another scenario to reproduce double-peaked radio light curves without introducing complicated setups on SN ejecta and CSM. The critical point is to take into consideration all of the possible shapes of the synchrotron spectra. The synchrotron spectrum can change its own shape depending on whether the electron spectrum is in the so-called "fast cooling" or "slow cooling" regimes, as well as on efficiencies of particle acceleration and magnetic field amplification. This idea has been already introduced in the modeling of gamma-ray burst (GRB) afterglows (Sari et al. 1998; Fan & Piran 2008; van Eerten et al. 2010; Ryan et al. 2020; Sato et al. 2021). We have applied the same framework to the modeling of radio SNe, incorporating synchrotron self-absorption into the calculation.

We show that a double-peaked radio light curve appears when the system becomes optically thin to synchrotron self-absorption at the observational frequency during the fast cooling regime. The first peak is characterized by the unity of the optical depth during the fast cooling regime, while the second peak appears when the synchrotron frequency related to the minimum Lorentz factor of electrons decreases down to observational frequency during the slow cooling regime. The observability of the double peak depends on the timescale of the transition from the fast cooling regime to the slow cooling regime. If the transition happens within a typical observational timescale of SNe, current observational instruments can potentially seize both peaks. This is fulfilled when the spectral energy distribution of nonthermal electrons is separated from the thermal distribution. This originates from the situation where (1) the SN shock transfers a large fraction of its kinetic energy into relativistic electrons, and (2) only a small population of electrons gets involved in shock acceleration. Our framework can thus provide a new clue to understanding the properties of shock acceleration physics in SNe.

This paper is organized as follows. Section 2 illustrates the framework and method of our model, explaining how to compute the luminosity of synchrotron emission from core-collapse SNe. Section 3 demonstrates our calculation, showing that radio light curves can take two types of morphology depending on electron acceleration efficiencies: a single-peaked and a double-peaked light curve. The interpretation of each solution is also described. Section 4 depicts the condition for the emergence and the observability of the double peaks in the radio light curve. Section 5 discusses implications for astrophysical phenomena and for shock acceleration physics. Finally, Section 6 summarizes this paper.

2.1. Concept

This study places emphasis on the evaluation of two characteristic Lorentz factors in a nonthermal electron's spectral energy distribution (SED). One is γ_m , defined as the minimum Lorentz factor of electrons injected by the SN shock. The other is γ_c , denoting the cooling break above which synchrotron (or inverse Compton) cooling is faster than adiabatic cooling. Depending on the inequality of γ_m and γ_c , an electron SED can change its shape. An electron SED with γ_m < γ_c is referred to as a "slow cooling" regime, and that with γ_m > γ_c as a "fast cooling" regime (Sari et al. 1998). The regime of the electron SED can indeed affect the resultant synchrotron spectrum (see, e.g., Granot & Sari 2002). We describe the definitions of γ_m and γ_c and the function of the electron SED for each regime in Section 2.2.3.

Figure 2 illustrates the schematic pictures of synchrotron spectra treated in this study. In a synchrotron spectrum, the synchrotron frequencies corresponding to γ_m and γ_c serve as the bending points. We define these frequencies as ν_m and ν_c . Additionally, the frequency below which the optical depth to synchrotron self-absorption exceeds unity characterizes the shape of the spectrum, which we denote as ν_a . In total, three characteristic frequencies become important in a synchrotron spectrum.

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Previous studies of radio SNe have implicitly assumed that the peak luminosity is determined by the frequency ν_a (Case I). The synchrotron spectrum then follows the distribution proportional to ν^5/2 and ν^(1−p)/2 in the lower and higher frequency ranges than ν_a , respectively, where p is the spectral index of the electron SED at the injection (e.g., Rybicki & Lightman 1979; Chevalier 1998; Chevalier & Fransson 2006; Fransson & Björnsson 1998; Chevalier et al. 2006). Furthermore, the spectrum in the higher frequency range than ν_c follows the softened distribution proportional to ν^−p/2. This method has been used to interpret the observational data of radio SNe (e.g., Soderberg et al. 2006; Maeda 2012, 2013a; Terreran et al. 2019).

However, it should be remarked that this framework requires the assumption that ν_m must be smaller than ν_a . ⁶ If the inequality among these three frequencies changes into ν_a < ν_m < ν_c , then the peak luminosity would be characterized by ν_m and the synchrotron SED in the frequency range ν_a < ν < ν_m becomes proportional to ν^1/3 (Case II). Furthermore when the inequality ν_a < ν_c < ν_m is realized, then the peak luminosity is characterized by the cooling break frequency ν_c and the spectrum in the frequency range ν_c < ν < ν_m should be proportional to ν^−1/2 (Case III). In the modeling of GRB afterglows, the spectra in Case II and Case III are also referred to as "slow cooling" and "fast cooling" regime, respectively (Sari et al. 1998). The radio light curve at a given frequency (ν_obs) is determined by which synchrotron spectrum is realized, and by the magnitude of ν_obs compared to the three characteristic frequencies.

This study addresses all of the synchrotron spectra shown in Cases I, II, and III. The critical difference from previous models for radio SNe is to incorporate the quantitative evaluation of γ_m and ν_m (however, see also Nakauchi et al. 2015). Depending on the inequality among ν_m , ν_a , and ν_c , the shape of synchrotron spectra and even the behavior of radio light curves can differ from those in previous studies. Hereafter, we describe our procedures to calculate the luminosity of synchrotron emission in SNe on the basis of this concept. We note that the synchrotron spectra discussed in this section are restricted in the case of ν_a < ν_c . It is possible that the situation of the inverse relation (ν_c < ν_a ) is realized depending on the input parameters or setup for the hydrodynamical configuration, which we do not delve into in the detailed discussion.

2.2. Methods

2.2.1. Hydrodynamics

We consider the situation in which synchrotron emission from SNe is generated at the interaction region between the SN shock and the CSM. This framework has been adopted as the standard model for the observed radio SNe (e.g., Chevalier 1998; Fransson & Björnsson 1998; Chevalier et al. 2006; Chevalier & Fransson 2006). We assume that the SN ejecta is undergoing homologous expansion and its density structure consists of the outer part given by the power-law distribution with the velocity coordinate (v) and the inner part with the flat density. The density structure of the outer part is described as follows:

$$\begin{eqnarray}&&{\rho }_{\mathrm{ej}}=\displaystyle \frac{3{M}_{\mathrm{ej}}}{4\pi {v}_{c}^{3}}\displaystyle \frac{n-3}{n}{t}^{-3}{\left(\displaystyle \frac{v}{{v}_{c}}\right)}^{-n},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{v}_{c}={\left(\displaystyle \frac{10{E}_{\mathrm{ej}}}{3{M}_{\mathrm{ej}}}\displaystyle \frac{n-5}{n-3}\right)}^{1/2},\end{eqnarray} \tag{ 2 }$$

where E_ej and M_ej are the kinetic energy and the mass of the ejecta, respectively (for the generalized formula, see, e.g., Chevalier & Soker 1989; Matzner & McKee 1999; Moriya et al. 2013). The index n determines the density gradient of the ejecta and characterizes the propagation of the SN shock. As for the CSM structure, we consider the power-law distribution of the CSM density with the index s as follows:

$$\begin{eqnarray}&&{\rho }_{\mathrm{CSM}}={ \mathcal D }{r}^{-s}=5\times {10}^{-19}{\tilde{A}}_{* }{\left(\displaystyle \frac{r}{{10}^{15}\,\mathrm{cm}}\right)}^{-s}\,{\rm{g}}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 3 }$$

Note that it is normalized at the CSM length scale of 10¹⁵ cm. Thus, when we assume the CSM velocity of 1000 km s⁻¹, ${\tilde{A}}_{* }=1$ represents a mass-loss episode with the rate of 10⁻⁵ M_⊙ yr⁻¹ occurring 0.3 yr before the explosion. For the case of s = 2, ${\rho }_{\mathrm{CSM}}=5\times {10}^{11}{\tilde{A}}_{* }{r}^{-2}$ is derived and then ${\tilde{A}}_{* }$ becomes equivalent with the notation A_* used in the literature (Maeda 2012; Suzuki & Maeda 2018).

Given the density profiles of the ejecta and the CSM, we can write down the time evolution of the SN shock. We employ the power-law solution derived in Chevalier (1982a, 1982b) as follows:

$$\begin{eqnarray}&&{R}_{\mathrm{sh}}={\left[\displaystyle \frac{(3-s)(4-s)}{(n-3)(n-4)}\displaystyle \frac{{u}_{c}^{n}}{{ \mathcal D }}\right]}^{\tfrac{1}{n-s}}{t}^{m},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{V}_{\mathrm{sh}}=\displaystyle \frac{{{dR}}_{\mathrm{sh}}}{{dt}}=\displaystyle \frac{n-3}{n-s}\displaystyle \frac{{R}_{\mathrm{sh}}}{t},\end{eqnarray} \tag{ 5 }$$

where m is a deceleration parameter given as m = (n − 3)/(n − s). The term u_c is defined as follows:

$$\begin{eqnarray}&&{u}_{c}={\left(\displaystyle \frac{3{M}_{\mathrm{ej}}}{4\pi {v}_{c}^{3-n}}\displaystyle \frac{n-3}{n}\right)}^{1/n}.\end{eqnarray} \tag{ 6 }$$

2.2.2. Particle Acceleration and Magnetic Field Amplification

It is believed that an SN shock drives acceleration of charged particles and amplification of a turbulent magnetic field during the propagation in the CSM (Bell 1978a, 1978b; Blandford & Ostriker 1978; Drury 1983). We follow the widely accepted parameterization (e.g., Chevalier & Fransson 2017), in which a fraction of the kinetic energy density dissipated at the SN shock is transported into nonthermal electrons and the magnetic field. We define the energy densities of nonthermal electrons (u_e ) and of the magnetic field (u_B ) as follows:

$$\begin{eqnarray}&&{u}_{e}={\epsilon }_{e}{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{u}_{B}=\displaystyle \frac{{B}^{2}}{8\pi }={\epsilon }_{B}{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2},\end{eqnarray} \tag{ 8 }$$

where ρ_sh = 9/8 × ρ_CSM(r = R_sh) is the shocked CSM density at the shock radius. ⁷ Here, we have introduced the parameters _e and _B that specify the energy transportation efficiencies. We note that there is still debate regarding the feasible values of these microphysics parameters (Caprioli et al. 2015; Matsumoto et al. 2015, 2017; Crumley et al. 2019).

2.2.3. Characteristic Lorentz Factors and an SED of Nonthermal Electrons

As mentioned in Section 2.1, an SED of nonthermal electrons is mainly characterized by two Lorentz factors: γ_m and γ_c . In the phenomenological perspective, γ_m can be estimated as follows (e.g., Sari et al. 1998; Ryan et al. 2020):

$$\begin{eqnarray}&&{\gamma }_{m}=\left(\displaystyle \frac{p-2}{p-1}\right)\displaystyle \frac{{\epsilon }_{e}}{{f}_{e}}\displaystyle \frac{{\mu }_{e}{m}_{p}{V}_{\mathrm{sh}}^{2}}{{m}_{e}{c}^{2}},\end{eqnarray} \tag{ 9 }$$

where m_e , m_p , c, and μ_e are electron mass, proton mass, light velocity, and the molecular weight of thermal electrons upstream, respectively. We have introduced a parameter f_e , the number fraction of nonthermal electrons compared to thermal electrons in the shocked CSM. By contrast, the cooling break Lorentz factor γ_c is determined by comparing the timescale contributed from synchrotron cooling (t_syn) and inverse Compton cooling (t_IC) with the adiabatic cooling timescale (t_ad). Namely, γ_c is given as a solution to the equation

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\mathrm{syn}}{t}_{\mathrm{IC}}}{{t}_{\mathrm{syn}}+{t}_{\mathrm{IC}}}={t}_{\mathrm{ad}}.\end{eqnarray} \tag{ 10 }$$

If we ignore the contribution from inverse Compton cooling, γ_c can be explicitly described as follows:

$$\begin{eqnarray}&&{\gamma }_{c}=\displaystyle \frac{6\pi {m}_{e}{cm}}{{\sigma }_{T}{B}^{2}t},\end{eqnarray} \tag{ 11 }$$

where σ_T is the cross section of Thomson scattering.

We note that the upper bound of the electron SED is characterized by the maximum Lorentz factor of accelerated electrons, which we denote as γ_M . It is basically determined by the balance between the acceleration timescale and synchrotron cooling timescale of electrons (e.g., Drury 1983; Kimura et al. 2020). We confirm that the typical SN parameters (e.g., V_sh ∼ 10⁹ cm s⁻¹, B ∼ 1 G) lead to γ_M ∼ 10⁶, which is sufficiently larger than γ_m as defined in Equation (9).

Nonthermal electrons attaining relativistic energies follow a single power-law distribution at the moment of the injection. We define the SED of relativistic electrons injected per unit time as C γ^−p . The SED is further modified by cooling processes, depending on which of γ_m and γ_c is larger (Sari et al. 1998; Fan & Piran 2008). In case of the fast cooling, γ_m > γ_c is realized and the electron SED is described as

$$\begin{eqnarray}N(\gamma )\simeq \left\{\begin{array}{cc}\displaystyle \frac{{{Ct}}_{\mathrm{syn}}(\gamma =1){\gamma }_{m}^{1-p}}{p-1}{\gamma }^{-2} & ({\gamma }_{c}\lt \gamma \lt {\gamma }_{m})\\ \displaystyle \frac{{{Ct}}_{\mathrm{syn}}(\gamma =1)}{p-1}{\gamma }^{-p-1} & ({\gamma }_{m}\lt \gamma ).\end{array}\right.\end{eqnarray} \tag{ 12 }$$

The low-energy component proportional to γ⁻² is attributed to the strong cooling process so that all of the injected electrons cool down to γ_c within the dynamical timescale. By constrast, in case of the slow cooling (γ_m < γ_c ), the SED can be given as follows:

$$\begin{eqnarray}N(\gamma )\simeq \left\{\begin{array}{cc}\displaystyle \frac{{{Ct}}_{\mathrm{ad}}}{p-1}{\gamma }^{-p} & ({\gamma }_{m}\lt \gamma \lt {\gamma }_{c})\\ \displaystyle \frac{{{Ct}}_{\mathrm{syn}}(\gamma =1)}{p-1}{\gamma }^{-p-1} & ({\gamma }_{c}\lt \gamma ).\end{array}\right.\end{eqnarray} \tag{ 13 }$$

In this regime, only the SED component more energetic than γ_c m_e c² experiences the softening due to the strong cooling. Combining Equations (12) and (13), the unified formula of the electron SED can be described as follows:

$$\begin{eqnarray}N(\gamma )=\left\{\begin{array}{cc}\displaystyle \frac{{{Ct}}_{\mathrm{cool}}(\gamma )}{(p-1)\gamma }{\left(\max (\gamma ,{\gamma }_{m})\right)}^{1-p} & (\gamma \gt \min ({\gamma }_{m},{\gamma }_{c}))\\ 0 & (\gamma \lt \min ({\gamma }_{m},{\gamma }_{c}))\end{array}\right.,\end{eqnarray} \tag{ 14 }$$

where t_cool is the total cooling timescale of electrons. The abovementioned descriptions of electron SEDs have been often employed in the modeling of GRB afterglows (e.g., Granot & Sari 2002; van Eerten et al. 2010; Ryan et al. 2020), including the synchrotron cooling. In radio SNe, two additional cooling processes can be important. One is inverse Compton scattering through which photons emitted from an SN photosphere would drain the energies of relativistic electrons (Fransson & Björnsson 1998; Chevalier et al. 2006; Chevalier & Franssono 2006; Maeda 2012). The other is Coulomb interaction that promotes energy exchange to thermal electrons in a dense circumstellar environment (Fransson & Björnsson 1998; Björnsson 2022). Then, the total cooling timescale is given by the reciprocal sum of each timescale as follows:

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}={\left(\displaystyle \frac{1}{{t}_{\mathrm{syn}}}+\displaystyle \frac{1}{{t}_{\mathrm{IC}}}+\displaystyle \frac{1}{{t}_{\mathrm{ad}}}+\displaystyle \frac{1}{{t}_{\mathrm{coulomb}}}\right)}^{-1},\end{eqnarray} \tag{ 15 }$$

and the four characteristic timescales are defined as follows:

$$\begin{eqnarray}&&{t}_{\mathrm{syn}}=\displaystyle \frac{6\pi {m}_{e}c}{{\sigma }_{T}}{\gamma }^{-1}{B}^{-2}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{IC}}=\displaystyle \frac{3\pi {m}_{e}{c}^{2}}{{\sigma }_{T}}{\gamma }^{-1}{L}_{\mathrm{bol}}^{-1}{R}_{\mathrm{sh}}^{2}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{ad}}=\displaystyle \frac{{R}_{\mathrm{sh}}}{{V}_{\mathrm{sh}}}=\displaystyle \frac{t}{m}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{coulomb}}=\displaystyle \frac{{m}_{p}{m}_{e}^{2}{c}^{3}}{4\pi {e}^{4}\mathrm{ln}{\rm{\Lambda }}}\gamma {\rho }_{\mathrm{sh}}^{-1},\end{eqnarray} \tag{ 19 }$$

where e, L_bol, and $\mathrm{ln}{\rm{\Lambda }}=30$ are the elementary charge, bolometric luminosity of the SN, and the coulomb logarithm, respectively (see also Matsuoka et al. 2019). We ignore the Klein–Nishina effect because the Lorentz factor of electrons responsible for radio SNe would be ∼1000 (see Section 3) and the energy of the seed photons is $\sim { \mathcal O }(1\,{\rm{eV}})$.

Finally, we give the definition of C as follows:

$$\begin{eqnarray}&&C=\displaystyle \frac{(p-2){u}_{e}}{{\gamma }_{m}^{2-p}{m}_{e}{c}^{2}}\displaystyle \frac{{V}_{\mathrm{sh}}}{{\rm{\Delta }}{R}_{\mathrm{sh}}},\end{eqnarray} \tag{ 20 }$$

where ΔR_sh is the geometrical thickness of the shocked CSM region. We assume ΔR_sh is proportional to R_sh, where the coefficient refers to the results in previous studies (Chevalier 1982a; Chevalier & Fransson 1994; Truelove & McKee 1999). Typically it falls within the range of ∼0.2–0.5, depending on the density gradient of the ejecta (n).

We note that when the CSM follows the steady wind configuration, there must be a phase when γ_m is larger than γ_c , with the electron SED following the fast cooling regime, toward t → 0. This is because γ_m decreases with time in our phenomenological definition while γ_c increases as a function of time. We refer readers to Appendix A, which specifies the proportional dependencies of the quantities including γ_m and γ_c .

2.2.4. Synchrotron Emission

Given the strength of the magnetic field and the profile of the nonthermal electrons' SED in the shocked CSM region, we can estimate the quantities related to synchrotron emission (e.g., Rybicki & Lightman 1979). The synchrotron spectrum emitted by one electron (P_syn(γ)) is described as a function of the observational frequency ν,

$$\begin{eqnarray}&&{P}_{\mathrm{syn}}(\gamma )=\displaystyle \frac{\sqrt{3}{e}^{3}B}{{m}_{e}{c}^{2}}G\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma )}\right),\end{eqnarray} \tag{ 21 }$$

where the following quantities are defined:

$$\begin{eqnarray}&&{\nu }_{\mathrm{syn}}(\gamma )=\displaystyle \frac{3{\gamma }^{2}{eB}}{4\pi {m}_{e}c},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&G(x)=\displaystyle \frac{1}{2}{\int }_{0}^{x}F\left(\displaystyle \frac{x}{\sin \theta }\right){\sin }^{2}\theta d\theta ,\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&F(x)=x{\int }_{x}^{\infty }{K}_{5/3}(y){dy}.\end{eqnarray} \tag{ 24 }$$

K_5/3(y) is the modified Bessel function with the order of 5/3. Here, the dependence on the pitch angle (θ) of emitter electrons (Rybicki & Lightman 1979) is absorbed into G(x) by taking the average over the pitch angle. We employ the fitting formula for G(x) suggested in Aharonian et al. (2010).

The emissivity (j_syn) and the absorption coefficient to synchrotron self-absorption (α_syn) are calculated as follows (Rybicki & Lightman 1979):

$$\begin{eqnarray}&&{j}_{\mathrm{syn}}=\displaystyle \frac{1}{4\pi }\int {P}_{\mathrm{syn}}(\gamma )N(\gamma )d\gamma ,\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\alpha }_{\mathrm{syn}}=\displaystyle \frac{-1}{8\pi {m}_{e}{\nu }^{2}}\int {P}_{\mathrm{syn}}(\gamma ){\gamma }^{2}\displaystyle \frac{\partial }{\partial \gamma }\left(\displaystyle \frac{N(\gamma )}{{\gamma }^{2}}\right)d\gamma .\end{eqnarray} \tag{ 26 }$$

Then, the optical depth to synchrotron self-absorption can be determined by

$$\begin{eqnarray}&&{\tau }_{\mathrm{syn}}={\alpha }_{\mathrm{syn}}{\rm{\Delta }}{R}_{\mathrm{sh}}.\end{eqnarray} \tag{ 27 }$$

Finally, the luminosity is calculated as follows:

$$\begin{eqnarray}&&{L}_{\nu }=4{\pi }^{2}{R}_{\mathrm{sh}}^{2}\displaystyle \frac{{j}_{\mathrm{syn}}}{{\alpha }_{\mathrm{syn}}}(1-{e}^{-{\tau }_{\mathrm{syn}}}).\end{eqnarray} \tag{ 28 }$$

In this section, we show examples of our radio light curve models. For simplicity, we parameterize only _e and f_e , which are related to the efficiency of electron acceleration. We fix the other parameters as follows: M_ej = 2 M_⊙, E_ej = 10⁵¹ erg, ${\tilde{A}}_{* }=50$, _B = 0.005, p = 3, s = 2, and n = 7. These values are in line with the typical parameter choice for stripped-envelope SNe (Matzner & McKee 1999; Chevalier & Fransson 2006; Lyman et al. 2016; Terreran et al. 2019; Maeda et al. 2021). In addition, we neglect the contribution from inverse Compton cooling for simplicity, as it depends on the time evolution of the bolometric luminosity of an SN itself.

The upper panels of Figure 3 and 4 show the radio light curves at the frequency of ν_obs = 5 GHz for the models with the parameter set of (_e , f_e ) = (10⁻², 10⁻²) and (10⁻¹, 10⁻³), respectively. We can obviously see that the light curves are different from each other, even though the parameters except for _e and f_e are the same. Indeed, these two parameters can affect the value of ${\gamma }_{m}(\propto {\epsilon }_{e}{f}_{e}^{-1})$ and the regime of the electron SED. This yields a diversity of the radio light curve morphology.

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Comparing the nonthermal electrons' SEDs between the two models allows us to investigate what gives rise to the difference. The lower panels in Figure 3 show the SEDs of nonthermal electrons at several ages for the model with (_e , f_e ) = (10⁻², 10⁻²). The corresponding synchrotron SEDs are displayed in Figure 5. The electron SED is in the fast cooling regime in the beginning phase (t = 0.7 day), but immediately evolves into the slow cooling regime (t = 3 days). In the early phase, the Lorentz factor responsible for the emission at the observational frequency ${\gamma }_{\mathrm{obs}}\simeq {(4\pi {m}_{e}c{\nu }_{\mathrm{obs}}/3{eB})}^{1/2}$ is smaller than γ_m , and the radio luminosity is mainly covered by electrons with γ ∼ γ_m . Due to the optically thick situation, the synchrotron emission follows the distribution with the relationship of L_ν ∝ ν² (Case II). Once γ_m decreases below γ_obs, nonthermal electrons with γ ∼ γ_obs become the main emitters of synchrotron emission at 5 GHz (t = 20 days). The system is still optically thick at ν = ν_obs and L_ν is proportional to ν^5/2 (Case I). The peak luminosity is characterized by the unity of the optical depth. After the system becomes optically thin, the radio luminosity begins to decay with time (t = 300 days). Except for the initial phase, this behavior is compatible with the classical interpretation of observed radio SNe (e.g., Chevalier et al. 2006; Chevalier & Fransson 2006; Soderberg et al. 2012; Horesh et al. 2020).

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The situation changes in the double-peaked model, where the electron SED remains in the fast cooling regime for a longer timescale, as illustrated in the lower panels of Figure 4. The resultant synchrotron SEDs are displayed in Figure 6. As γ_obs is smaller than γ_c , ⁸ electrons with γ ∼ γ_c become the main emitters of synchrotron radiation at ν = ν_obs. The system is optically thick at this frequency, and the synchrotron SED takes the form in Case III (t = 10 days). However, the density of the CSM swept up by the shock decreases with time. Thus once the system becomes optically thin at ν = ν_obs, the radio emission turns into decay. This is the physical process characterizing the first peak in the double-peaked radio light curve.

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Since γ_c increases with time due to the weakening of the synchrotron cooling, γ_c meets up with γ_m soon after the first peak, and this moment characterizes the dip, the local minimum of the radio light curve (t = 29 days). After that, the electron spectrum enters the slow cooling regime. As γ_obs < γ_m is satisfied even immediately after the transition, electrons with γ ∼ γ_m would be the main emitters of synchrotron radiation. The important effect is that γ_m decreases with time, resulting in the rapid increase in the population of the emitter electrons. This affords the rebrightening of radio emission (t = 200 days). Once γ_obs ≃ γ_m is achieved, the second peak is shaped, and the synchrotron SED evolves into the spectrum in Case II. After the second peak, the emission is responsible for electrons with γ ∼ γ_obs, and since the density of the CSM continues to decline with time the emission results in decay from the second peak (t = 1500 days). We refer readers to Appendix B for the mathematical interpretation of the origin of the double peak. We also mention that the possibly similar double-peaked characteristic has appeared in the numerical modeling of Wei et al. (2023), where multiwavelength afterglow emission associated with a magnetar giant flare or a fast radio burst is calculated on the basis of synchrotron emission in a similar way to our study.

We note that the critical difference between the two models lies in the timing of the regime transition. In the single-peaked model, the transition occurs during the optically thick phase, while in the double-peaked model, it happens in the optically thin phase. This means that all of the peaks discussed in this section are shaped in qualitatively different ways from each other. In a single-peaked model, the moment of τ_syn = 1 during the slow cooling regime is important for the formation of the peak, whereas in a double-peaked model, the moments of τ_syn = 1 during the fast cooling regime and of γ_obs ≃ γ_m are critical. This also indicates that the difference between these two models would emerge even in the synchrotron spectra: Case I is realized in the single-peaked model, while Cases II and III would be realized in the double-peaked model.

In the previous section we showed that, depending on the particle acceleration efficiencies, radio emission from SNe can reach the peak twice. It is useful to clarify what kinds of parameter sets lead to a double-peaked light curve. For this purpose, we define two characteristic timescales in radio light curves. One is t₁ when the system becomes optically thin to synchrotron self-absorption at ν = ν_obs. Timescale t₁ cannot be explicitly written down, but is derived as a solution of the equation τ_syn = 1.

Another timescale is t_*, at which the electron SED transitions from the fast cooling to slow cooling regime. When we neglect the cooling effect from inverse Compton scattering, t_* can be analytically described as follows:

$$\begin{eqnarray}&&{t}_{* }={\left[\left(\displaystyle \frac{p-2}{p-1}\right)\left(\displaystyle \frac{{\mu }_{e}{m}_{p}{\sigma }_{T}}{6\pi {m}_{e}^{2}{c}^{3}m}\right){V}_{0}^{2}{B}_{0}^{2}\displaystyle \frac{{\epsilon }_{e}{\epsilon }_{B}}{{f}_{e}}{\tilde{A}}_{* }^{\tfrac{n-4}{n-s}}\right]}^{\tfrac{1}{-4m+{ms}+3}},\end{eqnarray} \tag{ 29 }$$

where the shock velocity and the magnetic field are rewritten as ${V}_{\mathrm{sh}}={V}_{0}{\tilde{A}}_{* }^{\tfrac{-1}{n-s}}{t}^{m}$ and $B={B}_{0}{\epsilon }_{B}^{\tfrac{1}{2}}{{\tilde{A}}_{* }}^{\tfrac{n-2}{2(n-s)}}{t}^{\tfrac{m(2-s)}{2}-1}$, respectively (see also Appendix A). V₀ and B₀ stand for the coefficients of the shock velocity and the magnetic field strength that are connected to the CSM density, time, and _B . We note that t_* is independent of ν_obs unlike t₁.

The condition for the emergence of the double-peaked radio light curve is that the system should become optically thin at ν = ν_obs during the fast cooling regime. Namely, t₁ < t_* should be satisfied. Furthermore, in order to observationally capture the double-peak feature, t_* needs to be comparable to the typical observational timescale of SNe. In short, t_* ∼ t_obs is required, where t_obs ∼ 1–100 days is a typical observational timescale of radio SNe (Bietenholz et al. 2021). Otherwise, we would miss the moment of the regime transition. These two requirements are important for the emergence of the observable double peak in the radio light curve.

Figure 7 illustrates the parameter spaces that result in the observable double-peaked radio light curve at ν_obs = 5 GHz. Here, we focus on the dependencies of t_* on the CSM density and microphysics parameters _e , _B , and f_e . The other parameters (M_ej, E_ej, p, s, and n) are fixed to those in the double-peaked model in Section 3 to clearly show the parameter dependencies. We also plot the blue dashed lines to represent the critical parameter sets that cause the regime transition and the drop of the optical depth beneath unity simultaneously. This is determined as a solution to the equation of t₁ = t_*.

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The gray-shaded regions in Figure 7, where the double-peaked radio light curve can be observed, are obtained by the following conditions. First, to make the regime transition during the fast cooling we need to go into the parameter space below the critical line (t₁ < t_*). Second, we require t_* to be moderately long so that radio observations can trace the moment of the regime transition. From Figure 7, we can infer that a large _e and small f_e are preferred for the emergence of the observable double-peaked light curve. This parameter choice leads to high γ_m and corresponds to the situation where a substantial amount of energy from the shock is deposited into the limited population of nonthermal electrons. We also confirm that the emergence of the double peak prefers a gentle gradient of the SN ejecta (smaller n).

5.1. Reproduction of the Radio SN Data and Implications for Observations

Radio SNe observed so far have been fitted with the spectrum of Case I in Figure 2 (e.g., Chevalier 1998; Chevalier & Fransson 2006; Fransson & Björnsson 1998; Chevalier et al. 2006; Maeda 2012, 2013a; Terreran et al. 2019; Horesh et al. 2020). In this case, ν_m < ν_a < ν_c is implicitly assumed. By contrast, the model presented in this study can be regarded as the application of the modeling of GRB afterglows, where the synchrotron SED is determined depending on the inequality of characteristic synchrotron frequencies (see, e.g., Sari et al. 1998; Granot & Sari 2002; Ryan et al. 2020). This means that our method is applicable to all types of spectra (Cases I, II, and III), though we need to check whether our model can reproduce the observational data, as well as the consistency with previous studies.

Figure 8 shows the fitting results of the observed radio data of SN 2020oi and SN 2007bg based on our model. These models take into account the cooling timescale of inverse Compton scattering induced by seed photons from the SN photosphere. SN 2020oi is a Type Ic SN discovered in M100 (Rho et al. 2021; Gagliano et al. 2022), and radio observations have been conducted at the frequencies from 5 to 100 GHz (Horesh et al. 2020; Maeda et al. 2021). The observed radio light curve is characterized by a single peak. SN 2007bg is a broad-lined Type Ic SN (Young et al. 2010; Salas et al. 2013). Particularly, Salas et al. (2013) argued that the radio light curve in SN 2007bg exhibits the prominent double peak at 4.86 and 8.46 GHz, and proposed the two components of the CSM around the progenitor as the origin of the multiple peaks.

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As seen in Figure 8, our computations can successfully reproduce the models consistent with radio data of both the single-peaked radio SN 2020oi and the double-peaked radio SN 2007bg. We confirm that the synchrotron SED at the peak in SN 2020oi is fitted with the spectrum of Case I, demonstrating consistency with previous studies of radio SNe (Maeda et al. 2021). Furthermore, the model for SN 2007bg can reproduce the double-peaked radio light curve without invoking the multiple components of the CSM. Instead, the double-peaked feature is formed by the transition of the nonthermal electrons' spectrum from the fast cooling to the slow cooling regime. As expected, the model that fits the light curve of SN 2007bg has a high value of _e relative to f_e (_e ≃ 5 × 10⁻², f_e ≃ 10⁻⁴). We suggest that our model serves as an alternative explanation of the double-peaked radio light curve.

We note that SN 2007bg may be a special case of double-peaked radio SNe fitted by our model. As seen in Appendix B, our model reproduces the luminosity variation only by a factor of a few because the luminosity can be described by the power-law terms. Thus, the regime transition of the electron SED may be capable of explaining double-peaked radio SNe with a luminosity variation within a factor of a few. That said, it is also known that there are some radio SNe exhibiting a rebrightening by orders of magnitude (e.g., Margutti et al. 2017 for SN 2014C, and Maeda et al. 2023b for SN 2018ivc). We suppose that these SNe are beyond the explanation provided by our model, and therefore other physical processes may be important; an inhomogeneous CSM density structure (Maeda et al. 2021, 2023a) or time-dependent microphysical parameters (Huang & Li 2023) would be feasible candidates.

We suggest that multiwavelength observations can be diagnostics for the physical origin of the luminous radio emission seen in the late phase of SNe. For example, the existence of a massive CSM shell is supported in the case where the SN experiences the transformation of its morphology from a stripped-envelope SN to an interacting SN in optical wavelengths (e.g., Margutti et al. 2017). In this case, X-ray emission is also expected to follow. We suggest that multiband observations during the radio rebrightening phase would provide unique diagnostics for the physical mechanism producing the double peak in the radio emission.

Furthermore, we expect that multiband radio observations to measure the spectral index can potentially distinguish the physical origin of the peaks. As mentioned in Section 3, all of the light-curve peaks treated in this study appear in different ways with each other. The measured spectral index can be used to diagnose the morphology of the synchrotron spectra (see Figure 2) such that we could examine the occurrence of the regime transition of the electron SED.

5.2. Implications for Shock Acceleration Physics

5.3. Comparison with GRB Afterglows

5.4. Weakening the Dip Feature

In this study, we construct a model for calculating synchrotron emission induced by the interaction between SN ejecta and the CSM, where an electron SED is given by a broken power-law distribution. Therein two critical Lorentz factors become important: γ_m , the minimum Lorentz factor of the injected electrons, and γ_c , the Lorentz factor at which synchrotron (and inverse Compton) cooling rate balances with the adiabatic cooling rate. Depending on the inequality between γ_m and γ_c , the electron SED can take either the fast cooling or the slow cooling regime. This method has been usually employed in the context of modeling GRB afterglows, and we apply the framework to radio SNe in a similar way.

This generalization of radio SN modeling leads to the finding that an SN can show not only a usual single-peaked but also a double-peaked light curve in the radio synchrotron emission, even for single power-law CSM distribution, depending on the energy and number density of nonthermal electrons; this is a new regime that was not recognized previously. The first peak is characterized by τ_syn = 1 during the fast cooling regime, while the second peak is formed at the moment of γ_obs = γ_m after the transition from the fast cooling to the slow cooling regime. The condition for the emergence of the double-peaked radio light curve is that the system should become optically thin at the observational frequency during the fast cooling regime (t₁ < t_*); otherwise, only a single peak appears. Furthermore, to capture the double-peak feature observationally, the transition timescale from fast cooling to slow cooling regimes (t_*) should be comparable to typical observational timescales. The combination of high _e and low f_e is likely to satisfy these conditions. This corresponds to the situation where the nonthermal electrons' SED is separated from the thermal electrons' component. A broad-lined Type Ic SN 2007bg would be a possible double-peaked radio SN that can be fitted by our model. Our model can serve as a potential diagnostic for plasma physics related to shock acceleration mechanism in SNe.

The authors acknowledge Kohta Murase, Kaori Obayashi, Kenta Hotokezaka, Akihiro Suzuki, Toshikazu Shigeyama, Christopher Irwin, Yudai Suwa, and the referee for their fruitful comments. This study is supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant Nos. 20H01904, 21J12145 (TM), 22K14028, 23H04899 (SSK), 20H00174 (KM), 19H00694, 20H00158, 20H00179, 21H04997, 23H00127, 23H04894, and 23H05432 (MT). This research is supported by the National Science and Technology Council, Taiwan under grant No. MOST 110-2112-M-001-068-MY3 and the Academia Sinica, Taiwan under a career development award under grant No. AS-CDA-111-M04. S.S.K. acknowledges the support by the Tohoku Initiative for Fostering Global Researchers for Interdisciplinary Sciences (TI-FRIS) of MEXT's Strategic Professional Development Program for Young Researchers. M.T. acknowledges the support by the Japan Science and Technology Agency (JST) Fusion Oriented Research for Disruptive Science and Technology (FOREST) Program grant No. JPMJFR212Y. A part of the computations shown in this paper has been carried out in the computational cluster Yukawa-21 equipped in the Yukawa Institute for Theoretical Physics.

For completeness, we show the proportional dependencies of the quantities on SN age, CSM density, and shock acceleration efficiencies. They help us understand the behavior of the electron SED and the resultant radio light curve.

$$\begin{eqnarray}&&{R}_{\mathrm{sh}}\propto {\tilde{A}}_{* }^{\tfrac{-1}{n-s}}{t}^{m},\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&{V}_{\mathrm{sh}}\propto {\tilde{A}}_{* }^{\tfrac{-1}{n-s}}{t}^{m-1},\end{eqnarray} \tag{ A2 }$$

$$\begin{eqnarray}&&{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2}\propto {\tilde{A}}_{* }^{\tfrac{n-2}{n-s}}{t}^{m(2-s)-2},\end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray}&&B\propto {\left({\epsilon }_{B}{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2}\right)}^{1/2}\propto {\epsilon }_{B}^{\tfrac{1}{2}}{\tilde{A}}_{* }^{\tfrac{n-2}{2(n-s)}}{t}^{m(2-s)/2-1},\end{eqnarray} \tag{ A4 }$$

$$\begin{eqnarray}&&{\gamma }_{m}\propto {\epsilon }_{e}{f}_{e}^{-1}{V}_{\mathrm{sh}}^{2}\propto {\epsilon }_{e}{f}_{e}^{-1}{\tilde{A}}_{* }^{\tfrac{-2}{n-s}}{t}^{2(m-1)},\end{eqnarray} \tag{ A5 }$$

$$\begin{eqnarray}&&{\gamma }_{c}\propto {t}^{-1}{B}^{-2}\propto {\epsilon }_{B}^{-1}{\tilde{A}}_{* }^{\tfrac{2-n}{n-s}}{t}^{-m(2-s)+1},\end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray}&&{\gamma }_{\mathrm{obs}}\propto {B}^{-1/2}\propto {\epsilon }_{B}^{-1/4}{\tilde{A}}_{* }^{\tfrac{-n+2}{n-s}}{t}^{-m(2-s)/4+1/2},\end{eqnarray} \tag{ A7 }$$

$$\begin{eqnarray}&&\begin{array}{l}C\propto {\epsilon }_{e}{\rho }_{\mathrm{sh}}{V}_{\mathrm{sh}}^{2}{\gamma }_{m}^{p-2}{V}_{\mathrm{sh}}{\rm{\Delta }}{R}_{\mathrm{sh}}^{-1}\\ \propto {\epsilon }_{e}^{p-1}{f}_{e}^{2-p}{\tilde{A}}_{* }^{\tfrac{n-2}{n-s}-\tfrac{2(p-2)}{n-s}}{t}^{m(2-s)-3+2(m-1)(p-2)}.\end{array}\end{eqnarray} \tag{ A8 }$$

In this section, we derive the analytic formulae describing double-peaked radio light curves. The critical points are that (1) we should not adopt the δ-function approximation for the synchrotron emission profile in Equation (21) but should take the width of the profile into consideration, and (2) we need to take care of the treatments on the upper and lower bounds of the integrals in Equation (25).

Let us assume the optically thin situation at the observational frequency (ν) and start the integral of the emissivity (Equation (25)). To avoid the loss of generality, we consider a single power-law distribution for nonthermal electrons given as N(γ) = N_coeff γ^−k with the domain of the Lorentz factor defined as γ_a < γ < γ_b . If we assume a δ-function profile for the synchrotron emission, G(x) = g_δ δ(x − 1), then the emissivity would become

$$\begin{eqnarray}\begin{array}{rcl}{j}_{\mathrm{syn}} & = & \displaystyle \frac{\sqrt{3}{e}^{3}B}{8\pi {m}_{e}{c}^{2}}{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{(1-k)/2}\\ & & \times {\displaystyle \int }_{{x}_{b}}^{{x}_{a}}{N}_{\mathrm{coeff}}{g}_{\delta }\delta (x-1){x}^{(k-3)/2}{dx},\end{array}\end{eqnarray} \tag{ B1 }$$

where x = ν/ν_syn(γ) is defined, and x_a and x_b are the forms of x in which γ_a and γ_b are substituted, respectively. Terms x_a and x_b will be applied to one of x_c = ν/ν_syn(γ_c ), x_m = ν/ν_syn(γ_m ), or x_M = ν/ν_syn(γ_M ) depending on the function of the electron SED. The above integration cannot be appropriately evaluated unless x_b < 1 < x_a , equivalent to γ_a < γ_obs < γ_b . This is not physically reasonable, because even in the case of ${\gamma }_{\mathrm{obs}}\lt \min ({\gamma }_{m},{\gamma }_{c})$, electrons with Lorentz factors lying around $\min ({\gamma }_{m},{\gamma }_{c})$ mainly cover the radio luminosity at the observed frequency with its spectrum proportional to ν^1/3.

To derive a reasonable formula for the emissivity, we need to adopt the synchrotron profile with a finite width in Equation (21). Aharonian et al. (2010) proposed the strict expression of G(x) without using the integral but with the modified Bessel functions with orders of 1/3 and 2/3 as follows:

$$\begin{eqnarray}&&G(x)=\displaystyle \frac{x}{20}\left[(8+3{x}^{2}){\left({\kappa }_{1/3}\right)}^{2}+x{\kappa }_{2/3}(2{\kappa }_{1/3}-3x{\kappa }_{2/3})\right],\end{eqnarray} \tag{ B2 }$$

where κ_d = K_d (x/2) is defined. To obtain analytic expressions we consider the approximated formulae for the modified Bessel functions in the limit of x → 0 and x → ∞, which can be described as follows:

$$\begin{eqnarray}{\kappa }_{d}=\left\{\begin{array}{cc}\displaystyle \frac{{4}^{d}\pi }{\sqrt{3}{\rm{\Gamma }}(1-d)}{x}^{-d} & (x\to 0)\\ \sqrt{\displaystyle \frac{\pi }{x}}\left(1+\displaystyle \frac{1}{x}\displaystyle \frac{{\rm{\Gamma }}(d+3/2)}{{\rm{\Gamma }}(d-1/2)}+\displaystyle \frac{1}{{x}^{2}}\displaystyle \frac{{\rm{\Gamma }}(d+5/2)}{{\rm{\Gamma }}(d-3/2)}\right)\exp (-x/2) & (x\to \infty ),\end{array}\right.\end{eqnarray} \tag{ B3 }$$

where Γ(d) is the gamma function (Abramowitz & Stegun 1965). As for the limiting case of x → ∞, we have expanded the modified Bessel functions up to the second order to obtain the approximated expression with the minimum order of G(x). We obtain G(x) ≃ 1.808x^1/3 in the limit of x → 0, while we can deduce G(x) ≃ (179π/288)e^−x ≃ 1.952e^−x in the limit of x → ∞. Incorporating these two limiting cases, we assume

$$\begin{eqnarray}&&G(x)={{gx}}^{1/3}\exp (-x),\end{eqnarray} \tag{ B4 }$$

where g is a coefficient factor of 1.5 ∼ 2. The choice of g would affect the normalization of the radio luminosity but not alter the qualitative behaviors of the light curve. Equation (B4) is compatible with the formula written in Fan & Piran (2008).

So far we have considered the nonthermal electrons' SED given by a single power-law function. However, what we have discussed in this study is the broken power-law distribution realized in the fast cooling or slow cooling regime. Therefore, we need to take a summation of the two power-law distributions in the electron spectrum. When we consider the fast cooling case, the emissivity can be written as

$$\begin{eqnarray}&&\begin{array}{l}{j}_{\mathrm{syn}}\simeq \displaystyle \frac{\sqrt{3}{e}^{3}{Bg}}{8\pi {m}_{e}{c}^{2}}\left\{{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{-1/2}{\displaystyle \int }_{{x}_{m}}^{{x}_{c}}\displaystyle \frac{{{Ct}}_{\mathrm{syn}}(\gamma =1){\gamma }_{m}^{1-p}}{p-1}{x}^{-1/6}\exp (-x){dx}\right.\\ \left.+{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{-p/2}{\displaystyle \int }_{{x}_{M}}^{{x}_{m}}\displaystyle \frac{{{Ct}}_{\mathrm{syn}}(\gamma =1)}{p-1}{x}^{(3p-4)/6}\exp (-x){dx}\right\}.\end{array}\end{eqnarray} \tag{ B5 }$$

As discussed in Section 2.2.3, γ_M is orders of magnitude greater than γ_m so that we can approximate x_M ≃ 0. However, we are considering time-variable γ_m and γ_c changing their size relationship with each other. Therefore, we need to leave the upper and lower bound of the above integral finite, not to approximate x_m and x_c as zero. Instead, it is permitted to approximate the exponential factor in the integral as unity, $\exp (-x)\sim 1$, because x_m and x_c are now sufficiently smaller than 1. Given the above conditions, some algebra allows us to derive the following expression,

$$\begin{eqnarray}&&{j}_{\mathrm{syn}}\simeq \displaystyle \frac{3\sqrt{3}{e}^{3}{{BCt}}_{\mathrm{syn}}(\gamma =1){\gamma }_{m}^{1-p}g}{20\pi {m}_{e}{c}^{2}(p-1)}{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{1/3}\left({\gamma }_{c}^{-5/3}-\displaystyle \frac{3p-3}{3p+2}{\gamma }_{m}^{-5/3}\right).\end{eqnarray} \tag{ B6 }$$

We can see that the time evolution of the emissivity consists of two components: one is the term dependent on ${\gamma }_{c}^{-5/3}$, accounting for the effect that the radio luminosity is covered by electrons with Lorentz factors γ ∼ γ_c . The other is the term proportional to ${\gamma }_{m}^{-5/3}$, meaning that the decrease in γ_m leads to the reduction of the population of emitter electrons.

From Equation (B6), we can infer the origin of the local minimum of the double-peaked radio light curve. Multiplying this equation by $4{\pi }^{2}{R}_{\mathrm{sh}}^{2}{\rm{\Delta }}{R}_{\mathrm{sh}}$ leads to the formula of the radio luminosity, which can be described as follows:

$$\begin{eqnarray}&&{L}_{\nu }\simeq \displaystyle \frac{3\sqrt{3}\pi {{ge}}^{3}{R}_{\mathrm{sh}}^{2}{\rm{\Delta }}{R}_{\mathrm{sh}}{{BCt}}_{\mathrm{syn}}(\gamma =1){\gamma }_{m}^{1-p}}{5{m}_{e}{c}^{2}(p-1)}{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{\tfrac{1}{3}}\left({\gamma }_{c}^{-\tfrac{5}{3}}-\displaystyle \frac{3p-3}{3p+2}{\gamma }_{m}^{-\tfrac{5}{3}}\right)\end{eqnarray} \tag{ B7 }$$

$$\begin{eqnarray}&&\equiv {\nu }^{\tfrac{1}{3}}\left({L}_{1,c}{t}^{\tfrac{15m-6{ms}-4}{3}}-{L}_{1,m}{t}^{\tfrac{-5m-{ms}+11}{3}}\right),\end{eqnarray} \tag{ B8 }$$

where L_1,c and L_1,m are the coefficients independent of t and ν, originating from ${\gamma }_{c}^{-5/3}$ and ${\gamma }_{m}^{-5/3}$, respectively. As for the parameter set in Section 3, s = 2 and m = 0.8, and thus the index of the first term is −8/15 ≃ − 0.53. Because γ_c increases with time, the optically thin emission would decline with time intrinsically as long as γ_obs < γ_c . Furthermore, the second term depends on t^1.8, indicating that the decrease of γ_m brings about the suppression of the radio luminosity. In the end, the system reaches the situation in which γ_c and γ_m coincide with each other. This moment can be observed as the local minimum of the radio luminosity.

After the transition from the fast cooling to slow cooling regime, γ_m < γ_c is realized, and the shape of the electron SED changes. As for the epoch before reaching the second peak, we can write down the emissivity as follows:

$$\begin{eqnarray}\begin{array}{rcl}{j}_{\mathrm{syn}} & \simeq & \displaystyle \frac{\sqrt{3}{e}^{3}{Bg}}{8\pi {m}_{e}{c}^{2}}\left\{{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{(1-p)/2}{\displaystyle \int }_{{x}_{c}}^{{x}_{m}}\displaystyle \frac{{{Ct}}_{\mathrm{ad}}}{p-1}{x}^{(3p-7)/6}\exp (-x){dx}\right.\\ & & \qquad \left.+{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{-p/2}{\displaystyle \int }_{{x}_{M}}^{{x}_{c}}\displaystyle \frac{{{Ct}}_{\mathrm{syn}}(\gamma =1)}{p-1}{x}^{(3p-4)/6}\exp (-x){dx}\right\}\end{array}\end{eqnarray} \tag{ B9 }$$

$$\begin{eqnarray}&&\simeq \displaystyle \frac{3\sqrt{3}{e}^{3}{{BCt}}_{\mathrm{ad}}g}{4(3p-1)\pi {m}_{e}{c}^{2}(p-1)}{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{1/3}\left({\gamma }_{m}^{-p+1/3}-\displaystyle \frac{3}{3p+2}{\gamma }_{c}^{-p+1/3}\right),\end{eqnarray} \tag{ B10 }$$

where $\exp (-x)\sim 1$ is taken into consideration because γ_obs > γ_m is still satisfied immediately after the regime transition. Then, the radio luminosity has dependencies on time and observational frequency as follows:

$$\begin{eqnarray}&&{L}_{\nu }\equiv {\nu }^{\tfrac{1}{3}}({L}_{2,m}{t}^{\tfrac{7m-4{ms}+2}{3}}-{L}_{2,c}{t}^{m-{ms}+\tfrac{5}{3}+p(4m-{ms}-3)}),\end{eqnarray} \tag{ B11 }$$

where L_2,m and L_2,c are the coefficients independent of t and ν. The first and second terms originate from ${\gamma }_{m}^{-p+1/3}$ and ${\gamma }_{c}^{-p+1/3}$ in Equation (B10) and are proportional to t^0.4 and t^−3.3 for the parameter set in Section 3, respectively. This indicates that the second brightening in the double-peaked radio light curve is attributed to the time evolution of γ_m approaching γ_obs, while the contribution from electrons with γ ∼ γ_c can be neglected.

Finally, we describe the behavior of the radio luminosity after the second peak. During this phase, γ_m decreases below γ_obs so that the approximation of $\exp (-x)\sim 1$ becomes invalid. Instead, we could approximate x_m ∼ ∞ and x_c ∼ 0 in the limit of t → ∞ as an asymptotic form in Equation (B9). This operation neglects the contribution from the SED component in γ > γ_c . Then,

$$\begin{eqnarray}&&{j}_{\mathrm{syn}}\simeq \displaystyle \frac{\sqrt{3}{e}^{3}{Bg}}{8\pi {m}_{e}{c}^{2}}{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{(1-p)/2}{\int }_{0}^{\infty }\displaystyle \frac{{{Ct}}_{\mathrm{ad}}}{p-1}{x}^{(3p-7)/6}\exp (-x){dx}\end{eqnarray} \tag{ B12 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{\sqrt{3}{e}^{3}{{BCt}}_{\mathrm{ad}}g}{8\pi {m}_{e}{c}^{2}(p-1)}{\rm{\Gamma }}\left(\displaystyle \frac{3p-1}{6}\right){\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{syn}}(\gamma =1)}\right)}^{(1-p)/2},\end{eqnarray} \tag{ B13 }$$

can be deduced. The radio luminosity has dependencies on time and observational frequency as follows:

$$\begin{eqnarray}&&{L}_{\nu }\propto {\nu }^{\tfrac{(1-p)}{2}}{t}^{-m-\tfrac{5}{4}s+4+p(2m-2-\tfrac{s}{4})},\end{eqnarray} \tag{ B14 }$$

which is proportional to t⁻² for the parameter set in Section 3. In summary, we can find out from Equations (B8), (B11), and (B14) the time dependence of the radio luminosity decaying from the first peak, rebrightening in the second peak, and declining from the second peak, respectively. These behaviors, in addition to the effect of synchrotron self-absorption in the early phase, would shape the double-peaked radio light curve.